Long Term Simulations of Space Charge and Beam LossObserved in the CERN Proton Synchrotron

G. Franchetti∗, I. Hofmann∗, M. Giovannozzi†, M. Martini† , and E. Métral†

∗GSI, D-64291 Darmstadt, Germany† CERN, CH-1211 Geneva 23, Switzerland

Abstract. Long term storage of high intensity beams with small loss is required in the FAIR [1] project at GSI as well as for JPARC [2]. In this paper we discuss that an important contribution to the loss in bunched beams can be explained it terms of particles trapped into lattice and space charge driven islands. Dedicated experiments at the CERN-Proton Synchrotron to confirm the theoretical model have shown the existence of an emittance growth dominated regime for working points sufficiently far from the lattice resonance, and of a beam loss dominated regime for tunes very close to the resonance. While the emittance growth dominated regime has been investigated in previous studies [3], we focus here on the beam loss dominated regime and compare simulation results with measurements made in the CERN-Proton Synchrotron ring.

INTRODUCTION

The trapping of particles into resonance induced phasespace islands has first been considered for single passagethrough a resonance. The efficiency of the process hasbeen investigated in [4, 5] and mechanisms for detrap-ping have been studied as well [6]. Applications of thetrapping of particles have been proposed as a methodto clean the beam from halo particles [7] and recentlyfor multi-turn extraction [8, 9]. Contrary to a "useful"controlled trapping/detrapping of particles, in the beamloss issue for high intensity we are dealing with uncon-trolled trapping/detrapping phenomena. In the SIS100[10], for example, storage for 1 second of a high inten-sity bunched beam (with 1012 U28+ and ∆Qx ≈ 0.2) ina nonlinear lattice with a loss level not exceeding 1% isrequested. The proposed mechanism is as follows: whenthe ring tune Qx0 is set close above a single nonlinearresonance, stable islands appears at a certain amplitudecorresponding to the space charge detuning. Contrary tothe single passage through a resonance, in which the is-land control was achieved by changing properly the ma-chine tune, in the high intensity bunch space charge isnow controlling the island position along the longitu-dinal axes. Islands have to be further out in the phasespace in the center of the bunch (z = 0) to compensate thestronger space charge detuning, whereas in the head/tail(z =±zmax) the island will be further in due to the weakerspace charge. Hence some particles will be periodicallycrossed by the lattice induced/space charge controlled is-lands. These particles then perform a periodic resonancecrossing at the synchrotron frequency with the associ-ated trapping-detrapping phenomena. The complexity ofthe single particle dynamics in the self-consistent field isconsiderable and so is computer simulation. A study of

trapping conditions has been presented at this workshop[11]. A first simplified study was based on a model witha frozen coasting beam with a Gaussian distribution anda parametrically modulated intensity [12]. Unfortunatelyfully self-consistent simulations are affected by the ar-tificial noise created by the PIC solvers [13]: long termsimulations (thousands of machine turns) may transferthe usually acceptable PIC noise into unphysical emit-tance growth or excessive halo formation. Therefore, wehave proposed a long-term simulation using an analyti-cal expression for the electric field for a frozen chargedistribution presented in detail in [3] and applied to theSIS100 in [14]. In order to check the validity of thesesimulations, a campaign of experiments was started in2002 at the CERN-PS in a CERN-GSI collaboration. Theaim of these measurements was two-fold. One goal wasto investigate the new mechanism of beam loss/emittanceincrease and provide experimental data for code com-parison. The second was the investigation of the Mon-tague resonance with a purpose of a code benchmarking[15, 16]. The first code comparison presented in [3] em-ployed a constant focusing lattice and was only able toconfirm the experimentally observed emittance growth,but not the measured beam loss. In the present paper weextend the simulation to the full AG focusing lattice ofthe CERN-PS in order to get a better match of exper-imental and simulation conditions. In this comparisonwe are referring to recent experimental data obtained in2003.

MEASUREMENTS

The measurements were all performed keeping the ki-netic energy at the injection value of 1.4 GeV. Bunchprofiles were first measured at 180 ms after injection

(before powering the octupoles). Profiles were found tobe Gaussian in both vertical and horizontal. The chro-maticity was close to the natural one, and the effect ofthe momentum spread of 3.3× 10−3 (at 2σ ) was foundto be 27% of the maximum space charge induced tune-spread ∆Qx = 0.075 for a particle with small amplitude.At 180 ms after injection two calibrated octupoles eachwith integrated strength of K3 = 0.6075× I m−3, withI octupole current, were powered to -20 A in order toexcite the resonance 4Qx0 = 25. The measurement win-dow was set to 1.2 s (4.4 × 105 turns) during whichthe bunch intensity was monitored with a current trans-former. At selected times we measured transverse beamprofiles with the flying wire (20 m/s), fitted them with aGaussian profile, and determined the corresponding rmsemittances. In most cases profiles were actually foundquite close to Gaussian. The vertical machine tune wasset to Qy0 = 6.12, and the horizontal tune was varied inthe interval 6.24 < Qx0 < 6.32 so as to link the octupoleresonance crossing with different positions of the spacecharge tune-spread. The parameters of the experimentare summarized in Table 1. In Fig. 1 we plot, as a func-

TABLE 1. Summary of experimental settingsParameter Value Units

Kinetic energy 1.4 GeVParticles per bunch 1×1012

Bunch length (4σ ) 180 nsEmittances (normalized at 2σ )∗ 25.5/10 mm mradMomentum spread (2σ ) 3.3×10−3

Derived maximum tuneshifts∗ 0.075/0.12PS circumference 628 mBeam pipe axes∗ 14/7 cmPS superperiodicity 10Octupole current -20 A

∗ horizontal/vertical

tion of Qx0, the emittances and beam intensity 1.2 s afterinjection. In the interval 6.28 < Qx0 < 6.32 we find thetypical emittance growth regime, which is characterizedby the maximum emittance growth of 42% at the tuneQx0 = 6.28. The beam loss regime, instead, is located inthe interval 6.25 < Qx0 < 6.28, and the maximum beamloss of ∼ 32% is found at Qx0 = 6.265. We also stud-ied the time dependence of the longitudinal bunch pro-file. These results shown here in a standard waterfall plot(Fig. 2a) suggest that the lost particles make the longitu-dinal distribution shrink in amplitude and size. In order toquantify this visual pattern we computed the time evolu-tion of the bunch length by performing a Gaussian fit foreach of the profiles of Fig. 2a. These results are plottedin Fig. 2b. In the same picture we plot also the integratedintensity of each longitudinal bunch profile for each timemeasurement. The two curves in Fig. 2b show that thereis a direct relation between beam loss and bunch short-ening. Unless an unexpected transverse-longitudinal cor-

relation takes place, Figs. 2 a,b suggest that lost particlesare not only those with large transverse amplitude, butalso with large synchrotron amplitude. This experimen-tal evidence is consistent with the condition of periodicresonance crossing. In fact particles with small longitu-dinal amplitude and small transverse amplitude will al-ways have an effective tune Qx sufficiently below 6.25,because their motion is confined in the denser region ofthe bunch, hence they will never be extracted. Particleswith large longitudinal amplitude, instead, can periodi-cally cross the resonance and therefore may get trappedand eventually get lost.

SIMULATION

3D simulations so far have been performed by using asimplified axisymmetric frozen bunch model with den-sity profile ρ ∝ (1− t)2 [17]. Here t = (r/R)2 +(z/Z)2,with r =

√

x2 + y2, R the transverse beam edge ra-dius and Z the longitudinal edge. This analytic frozenbunch model has the advantage that noise-free spacecharge forces are obtained. In the simplified axisym-metric model the horizontal and vertical axes were ar-tificially equal, therefore we have progressed to mod-eling ellipsoidal bunches [18]. For the simulations re-ported here we used an ellipsoidal Gaussian distributionρ ∝ exp(−t/2), with t = (x/σx)

2 + (y/σy)2 + (z/σz)

2,and σx,σy,σz arbitrary rms bunch sizes. The tracking isperformed by computing the average beam size in hor-izontal/vertical axes, then the space charge algorithm isinitialized keeping the 3 bunch axes frozen. This model-ing is neglecting the local transverse modulation of thebunch envelopes which we expect to have only a minorinfluence on the island position due to its rapid oscil-lation. A further improvement in simulations has beenreached by using the AG focusing PS lattice as provided

0.6

0.8

1

1.2

1.4

6.24 6.26 6.28 6.3 6.32Qx0

εx / εx0 Exp.

I / I0 Exp.

FIGURE 1. Experimental findings: normalized final emit-tance and beam intensity as function of the working point.

in [19], including the measured lattice nonlinearities asdiscussed in [20]. With this modeling the beam is nowexperiencing the correct nonlinear forces from the oc-tupole.

0.5

0.6

0.7

0.8

0.9

1

0.2 0.4 0.6 0.8 1 1.2time [s]

L / L0

I / I 0At

2+ Bt + C

A√

t + Bt + C

0

5

10

15

0 100 200 300 400bunch length [ns]

tim

e

x 10

0 [

ms]

a)

b)

FIGURE 2. Measurements at Qx0 = 6.265,Qy0 = 6.12: a)Waterfall plot of the longitudinal profile as function of time; b)Beam intensity and bunch length as function of storage time.The solid lines are fitted with data by using the functionaldependance reported.

Dynamic aperture and space charge

In previous work [3] the PS lattice was simulated ina constant focusing approximation. This simplified theparticle dynamics, but it also introduced enhanced hor-izontal and reduced vertical octupolar nonlinear forcesat the location of the external octupoles as in this caseβx = βy = 16 m. In contrast, by using the real PS lat-tice, the beta functions are βx = 11.7 m, βy = 22.2 m atthe location of the octupoles. It becomes then necessaryfirstly to investigate if beam loss may be attributed to anexcessive shrinking of the dynamic aperture. To exploreroughly the dynamic aperture of the PS we have carriedout a numerical test by searching the maximum stable

radius of test particles placed into 20 different directionsin the upper half of the x− y plane. In these calculationsspace charge is not included. The stability condition wassearched within 103 turns (short term dynamic aperture).We firstly computed the dynamic aperture without exter-nal octupole (I = 0 A), and found that it is at 9σ , for allthe tune range, where σ is the horizontal rms beam sizeof the injected beam. This value of the dynamic aperturesets the inner most part of the stable region practicallyon the beam pipe, i.e. the mechanical acceptance is allinside the stable region. By activating the octupoles withI = −20 A, the short term dynamic aperture is lowered to8σ for 6.27 < Qx0 < 6.32, but near Qx0 = 6.25 it shrinksto 3.5σ . This value is small enough to intercept the tailof a Gaussian distribution. However, a 2D multiparticlesimulation (without space charge) on the same time scalehas shown no particle loss. An upper bound to the beamloss can be obtained by cutting the 2D Gaussian distri-bution in energy at 3.5σ , i.e. keeping only particles suchthat εx,s/ε̃x + εy,s/ε̃y < 3.5, with εx/y,s the single particleemittances, and ε̃x/y beam rms emittances. This estimategives only 0.5% beam loss.

0.51

1.52

2.53

2.557.51012.515

6.2

6.22

6.24

6.26

zm

ax / σz

xmax / σx

Qx

FIGURE 3. Nonlinear tune Qx as function of the maximuminitial amplitude (xmax,zmax) for Qx0 = 6.265. The longitudinalmotion is frozen.A further inspection at 105 turns shows that close toQx0 = 6.25 the dynamic aperture without space chargeshrinks from 3.5σ to 3σ . A 2D simulation for thistimescale has shown beam loss of 0.1%. Again these re-sults suggest that the dynamic aperture alone as modeledhere and ignoring space charge cannot explain beam loss.We also note the significant difference between the studyof the single particle stability, where the minimum dy-namic aperture is located at Qx0 = 6.25, from the exper-imental finding of Fig. 1, where the maximum loss oc-curs at Qx0 = 6.265. In order to understand the originof this difference we studied a simplified constant focus-ing 2D model of the PS with the nonlinearities of onemagnet. In this model the longitudinal motion is kept

frozen. The aim of this model is to study the nonlineartune of a test particle as function of its initial coordinatesx = xmax,z = zmax,x′ = z′ = 0. The tune was computedwith an FFT in 4096 turns. In Fig. 3 we computed this"frequency map" [21]for the bare tune of Qx0 = 6.265.We see that for x ∼ z ∼ 0 the nonlinear tune Qx hasthe lower value Qx = Qx0 −∆Qx with ∆Qx = 0.075. Atthe longitudinal position of z = 3σz the space chargeis quite reduced and only the detuning due to the oc-tupole is present. Note that detuning of octupole andspace charge act in the same direction (they reduce thenominal tune) but in a different way: the space charge de-tuning is stronger on axis whereas the octupole detuningis stronger off axis as it can be seen at z = 3σz. The flat-top in the picture represents all the initial conditions ofparticles whose tune is locked on the 4th order resonance.It is approximately given where the space charge detun-ing matches the octupole induced resonance condition. Itis interesting to note that this flat-top does not reach theplane z = 0, but stops at z = 1 as there a chaotic regionis met. We also note that the flat-top reaches the maxi-mum extension of 5σ far below the dynamic aperture of10σ . However the chaotic region extends from 5σ to thedynamic aperture. This picture suggests that trapped par-ticles follow the flat-top, and when they reach the chaoticregion they can be brought to the dynamic aperture andget lost. This case occurs only at the tune Qx0 = 6.265.For comparison we have calculated the same "frequencymap" for Qx0 = 6.2685. In this case the flattop crosses theplane z = 0 at 5σ . This becomes consequently the maxi-mum extension of trapped particles which cannot be lost.For this tune only an emittance growth may occur.

3D simulations with synchrotron motion

Using the AG structure of the PS ring increases con-siderably the CPU-time required for simulations. For thisreason we have restricted the number of turns to 1.5×105. Following the procedure of the experiment we haveperformed simulations for the relevant working pointsused in the measurements. The initial bunch distributionwas Gaussian with 2000 macroparticles with transverseemittances εx = 25.5 mm-mrad, εy = 10 mm-mrad nor-malized at 2σ . Chromatic effects have been neglectedhere. In the simulations the PS beam pipe (14/7 cm)has been assumed constant throughout the ring, whichis a reasonable assumption. In Fig. 4 we show the resultof the simulations at 1.5× 105 turns and plot for con-venience also the final experimental data. The compari-son shows a similar pattern: an emittance growth regimefor larger tunes next to a loss dominated regime closerto Qx = 6.25. In the emittance growth regime the agree-ment between measured and simulated data, taken at thesame number of turns, is reasonably good. This is shown

0.5

0.75

1

1.25

1.5

1.75

2

6.24 6.26 6.28 6.3 6.32Qx0

εx / εx0 Sim.

I / I0 Sim.

εx / εx0 Exp.

I / I0 Exp.

FIGURE 4. Final beam emittance (solid) and intensities(dashed) versus working point in simulations (no markers) upto 1.5×105 turns and experiment (markers) up to 4.4×105 .

in Fig. 5 for the working point corresponding to the max-imum emittance growth in the experiment. The discrep-ancy is larger in the loss dominated regime, where thesimulation gives a maximum 8% for Qx0 = 6.26, whilethe maximum measured beam loss is found to be ≈ 25%at Qx0 = 6.265 both after 1.5× 105 turns. This suggeststhat we should re-examine all approximations made insimulations, but also the accuracy to which the nonlin-ear lattice used in the simulations represents the real ma-chine nonlinearities at the specific working points con-sidered here. In fact, we have also measured the loss atQx0 = 6.265 after 4.4× 105 turns with the octupole offand found ≈ 20%, compared with ≈ 30% for the oc-tupole on. The no-octupole measurement showed, how-ever, no emittance growth regime, which suggests thatthe natural lattice octupoles are too weak to cause effi-cient trapping/detrapping. Hence, the loss difference be-

1

1.05

1.1

1.15

1.2

0.2 0.3 0.4 0.5time [s]

ε x / ε

x0

at (

2σ)

mm

-mra

d

εx Exp. Qx0 = 6.285

εx Sim. Qx0 = 6.28

FIGURE 5. Comparison of emittance time evolution in mea-surements (markers) and simulation at Qx0 = 6.28.

tween octupole on/off, about 10%, should be mainly dueto trapping/detrapping by the additional octupole. Thisloss difference is in much better agreement with the sim-ulation result of maximum 8% loss with the octupoleon. In future work the effect of chromaticity should alsobe added as it enters directly into the resonance condi-tion and therefore into the trapping/detrapping probabil-ity as well as into the maximum halo radius. It shouldalso be noted that experimental beam loss of the orderof tens of percent leads this study into a realm, whereself-consistent simulation would be desirable.

0.9

0.92

0.94

0.96

0.98

1

0.2 0.3 0.4 0.5

I / I0

L / L0

time [s]

At2+ Bt + C

0.85

0.9

0.95

1

1.05

0.2 0.3 0.4 0.5time [s]

ε x / ε

x0 a

t (2

σ) m

m-m

rad

a)

b)

FIGURE 6. Simulation for Qx0 = 6.26: a) rms horizontalemittance versus time; b) beam intensity and bunch length withfitted curve as function of storage time.

We expect that with self-consistency the loss doesn’tsaturate, since with weakened space charge - by the loss- additional particles may cross the resonance. The smallloss in the simulation of Fig. 6 makes it unlikely that self-consistency would make a noticeable change. To furthersupport the interpretation of our findings we have stud-ied the correlation between beam loss and bunch short-ening, which is shown in Fig. 6. The bunch shorteningis comparable with the experimental results of Fig. 2bat 1.5× 105 turns, even if for a slightly different bare

tune (in fact the experimental beam loss data are rela-tively flat between Qx0 = 6.26 and Qx0 = 6.265). Notethat although the measured and simulated emittance andbunch length follow a similar evolution in the first 0.5seconds, the beam loss is quite different.

CONCLUSION

The experimental study in the CERN-Proton Syn-chrotron and comparison with simulations has reacheda sufficiently good agreement in the emittance growthregime. With respect to previous studies we now alsosucceeded to confirm the beam loss regime. There, theagreement with the measurement is also reasonablygood if we take the difference in measurements withoctupole on and off, which eliminates the pure dynamicaperture effect. Further steps will be needed to get a bet-ter quantitative code modelling of the machine dynamicaperture exactly at the working points used here. In orderto confirm that the mechanism proposed here is indeedaccelerator independent we are planning to carry out asimilar measurement campaign in the SIS18 at GSI.

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